We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3 and find two critical curves intersecting at one point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time on and below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.

中文翻译：

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退化的两个物种和交叉吸引系统中全球存在和爆炸的尖锐条件

我们考虑具有两个物种和两个刺激的退化趋化模型，维度 d ≥ 3，并找到两条临界曲线相交于一个点，将问题的弱解的全局存在和爆炸分开。更准确地说，在这些曲线之上（即亚临界情况），问题承认通过逼近系统的强解的极限获得的全局弱解。基于解的二阶矩，构建初始数据以确保在这些曲线上和下的有限时间内发生爆炸（即临界和超临界情况）。此外，在临界曲线上讨论了自由能泛函极小值的存在与否，如果初始数据规模较小，则解在时间上全局存在。我们还研究了临界线之间的交叉点，在该交叉点中，再次给出了关于群众的精炼标准，以区分全球存在和爆炸之间的二分法。我们还表明，两个物种的爆炸是同时发生的。